my $cookies = HTTP::Cookies->new();
my $ua = LWP::UserAgent->new( cookie_jar => {} );
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ICBW but it looks to me like you are setting up a new cookie jar in $cookies but then failing to associate that with your user agent. The docs for LWP::UserAgent also suggest using a HTTP::CookieJar::LWP instead. Try replacing these lines with

my $cookies = HTTP::CookieJar::LWP->new;
my $ua = LWP::UserAgent->new( cookie_jar => $cookies );
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and see if that gets you any further.

You're already using Rest::Client to fetch the initial cookie. You should inspect $client->responseHeaders to see where the cookie you want lives. Most likely, it is something like:

$client->POST('URL', ... credentials ...);
my $token = $client->responseHeader('Set-Cookie');
...
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A better approach is likely to use $ua for fetching the token instead, as that gives you far better control over the headers, and the $cookies cookie jar, if you properly associate it with LWP::UserAgent:

my $response = $ua->post('URL', Content => ... credentials ...);
my $token = $response->headers->{'Set-Cookie'}; # if there is only one
+ cookie line

my $token = $cookies->get_cookies( 'URL', 'AUTH_TOKEN' );

...
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